Picking Numbers for Variables in the Answers

Picking numbers works because an equivalence that holds for every value of n has to hold for the one value you choose too. So instead of wrestling with fractions and variables, you substitute n = 5, say, compute what the original expression equals, then plug n = 5 into each answer choice. Only the choice matching that number can be right, since a wrong choice will fail for at least some values of n, and you just need to catch it failing once.

The catch is which number you pick. If you choose n = 0 or n = 1, many different-looking expressions collapse to the same value: (n+1) and (n²+1) both equal 1 at n = 0, and both equal 2 at n = 1. Those numbers hide the differences you're trying to expose. A value like 2 or 5 keeps the expressions spread apart, so mismatches show up clearly. Negative numbers and fractions can also work, but small positive integers bigger than 1 are usually safest and fastest to compute with.

Sometimes two choices still tie after your first substitution. That doesn't mean both are right, it means your number wasn't strong enough to separate them. The fix is to pick a second, different number and test only the surviving choices against it. Whichever one matches both substitutions is the answer. This technique sits alongside working backward from answer choices in equations: both let you trade abstract manipulation for concrete checking, which is faster and less error-prone under pressure.